Separators of fat points in P^n

TL;DR

This paper generalizes the concept of separators for points in projective space to fat points with multiplicities, linking their minimal separator degrees to Hilbert functions and syzygies, thus extending classical results.

Contribution

It introduces a new definition of separators for fat points in P^n and relates their degrees to Hilbert functions and syzygy modules, generalizing known results.

Findings

The degree of minimal separators determines the Hilbert function of fat point schemes.

Entries of deg_Z(P_i) relate to shifts in the last syzygy module.

Knowing deg_Z(P_i) and the Hilbert function allows computation of related schemes.

Abstract

In this paper we extend the definition of a separator of a point P in P^n to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z = m_1P_1 + ... + m_iP_i + .... + m_sP_s in P^n and Z' = m_1P_1 + ... + (m_i-1)P_i + .... + m_sP_s. We associate to P_i a tuple of positive integers of length v = deg Z - deg Z'. We call this tuple the degree of the minimal separators of P_i of multiplicity m_i, and we denote it by deg_Z(P_i) = (d_1,...,d_v). We show that if one knows deg_Z(P_i) and the Hilbert function of Z, one will also know the Hilbert function of Z'. We also show that the entries of deg_Z(P_i) are related to the shifts in the last syzygy module of I_Z. Both results generalize well known results about reduced sets of points and their separators.